1977 IMO Problems/Problem 1
In the interior of a square we construct the equilateral triangles Prove that the midpoints of the four segments and the midpoints of the eight segments are the 12 vertices of a regular dodecagon.
Just use complex numbers, with , , and . With some calculations, we have , , and . Now it's an easy job to calculate the twelve midpoints and to find out they are all of the form , with , and the result follows.
The above solution was posted and copyrighted by Joao Pedro Santos. The original thread for this problem can be found here: 
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